Genus and crosscap of Normal subgroup based power graphs of finite groups

Abstract

Let $H$ be a normal subgroup of a group $G$. The normal subgroup based power graph ${\Gamma }_H(G)$ of $G$ is the simple undirected graph with vertex set $V({\Gamma }_H(G))=(G\setminus H)\cup \{e\}$ and two distinct vertices $a$ and $b$ are adjacent if either $aH=b^{m}H$ or $bH=a^{n}H$ for some $m,n\in \mathbb{N}$. In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs $(G,H)$, where $H$ is a non-trivial normal subgroup of $G$, such that the genus of ${\Gamma }_H(G)$ is at most $2$. Moreover, we determine all the subgroups $H$ and the quotient groups $\frac{G}{H}$ such that the cross-cap of ${\Gamma }_H(G)$ is at most three.